The Quasicontinuity of Delta-Fine Functions

Abstract

We wish to take advantage of the exploratory nature of the Inroads section to report on progress toward answering a question we posed with Richard O’Malley in [3]. There we noted the difficulty we were having trying to find an effective characterization of the class UPA of universally polygonally approximable functions. While several related subclasses of Baire one functions have aesthetically pleasing characterizations, UPA strikes us as more elusive. One difficulty is that it is not closed in the sup metric [3]. Thus, if one is looking for a geometric characterization, one should perhaps look, instead, for a characterization of its closure, UPA. In [3] we defined the class DF of delta-fine functions, which is closed, showed that UPA ⊆ DF, but were unable to determine if UPA = DF. Although the theorem presented in this paper doesn’t decide this question, it does provide additional insight into the similarity of these two function classes. In [2] we showed that the set of points at which a UPA function fails to be quasicontinuous is very small in the sense of porosity; indeed, it was shown to be σ-(1 − )-symmetrically porous for every > 0. (In [1] we examined how tantalizingly close this result is to being sharp.) Here we show that the same exceptional behavior is true for the class DF; that is, for every > 0 the set of nonquasicontinuity points for a delta-fine function is σ-(1− )-symmetrically porous. Although there are obvious similarities between the proof presented here and the UPA case proved in [2], the proofs differ at critical points. The main